Hopf fibration

The Hopf fibration is the map h: S³ → S² defined by h(z₁, z₂) = (2z₁z̄₂, |z₁|² − |z₂|²), where S³ is viewed as the unit sphere in ℂ² and S² as the unit sphere in ℝ³ ≅ ℂ × ℝ.¹ This map is a fiber bundle with fiber S¹, meaning that locally it looks like the projection S¹ × U → U, but globally it is nontrivial: the total space is S³, not S² × S¹.²

The Hopf fibration is the first example of a nontrivial fiber bundle and a generator of the homotopy group π₃(S²) ≅ ℤ.³ It also demonstrates that the higher homotopy groups of spheres can be nontrivial, contradicting the naive expectation that they might vanish. The Hopf map is the attaching map for the 4-cell in the CW complex of ℂP², and it induces a cup product structure in cohomology.⁴

Geometrically, the Hopf fibration can be visualized via stereographic projection: the fibers are great circles that link each other, and each pair of fibers is linked with linking number 1. The fibration is related to the quaternionic Hopf fibration S⁷ → S⁴ and the octonionic version S¹⁵ → S⁸, which are the only sphere fibrations with fiber S¹, S³, and S⁷.⁵

In physics, the Hopf fibration appears in the description of magnetic monopoles, where the Dirac monopole charge is quantized due to the nontrivial topology of the U(1) bundle. It also plays a role in quantum information theory, where the Bloch sphere represents the state space of a qubit, and the Hopf fibration describes the map from the state vector to the Bloch vector.

The Hopf fibration has several lesser-known facets. It is related to the concept of the Hopf invariant, which is a homotopy invariant that can be computed via the linking number of two fibers.³ The fibration also appears in the study of the geometry of the 3-sphere, where it gives a foliation by great circles, known as the Hopf foliation. In addition, the Hopf fibration is used in the construction of the Hopf map in the context of the Adams spectral sequence, and it is a key example in the theory of principal bundles.² A curious fact is that the Hopf fibration is the only nontrivial fiber bundle with fiber, base, and total space all spheres, as proven by Adams in 1960.⁵

Fiber bundle

A space that locally looks like a product of a base space and a fiber, but may be globally twisted.

Homotopy group

A group that captures the notion of higher-dimensional holes in a topological space.

Linking number

An integer that describes how two closed curves are intertwined in 3-dimensional space.

Stereographic projection

A mapping that projects a sphere onto a plane, preserving angles.